Don't you think some players start to care and try harder in the playoffs? Not really 'clutchness', but it can't be attributed to randomness either.

Maybe, but until someone can demonstrate that that is true, rather than simply assert that it true, it shouldn't be part of the analysis. All we know for a fact is that the same player plays in the regular season and in the playoffs. If you want to assert that there is something inherently and predictably different about this player's ability to perform based on what time of year that is, you need to provide evidence. Otherwise you fail to show anything that cannot be explained by normal variance in game-to-game performance.

Claude Lemieux scored .34 goals per game in the playoffs, and .31 in the regular season. He had some big playoff years, but also years where he scored 1 in 11, 3 in 19 and 4 in 23. When looking at players with clutch reputations, proponents tend to focus on the data that supports the assertion while downplaying that which does not. It's classic confirmation bias: I believe that clutch play exists, therefore I will find and present data to support it. It's starting with the conclusion and then looking for supporting evidence. You need to look at the evidence first, and derive your conclusion from that.

And then, moving beyond all that, is the underlying value judgement. If certain players try harder in the playoffs, that means they are not giving 100% in the regular season. Yet they are held up as heroes, while those who give the same effort year-round are not.